Introduction to Computational Origami pp 25-58 | Cite as

# Common Nets of Boxes

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## Abstract

In this chapter, we first consider problems that are easy to handle with computers. Since it seems to be hard to deal with the real coordinate system, polygons on a square grid would be reasonable. Speaking of polyhedra that can be folded from a polygon on a square gird, the first thing that comes to mind is a rectangular parallelepiped, or “box”. Is there a single polygon on a square grid that can be folded into multiple rectangular parallelepipeds? The answer is [Yes]. In 1999, Biedl et al. succeeded in actually finding out two polygons that satisfy the following property: “each of two can fold into two different types of boxes” [BCD+99]. (The actual examples are shown in [DO07, Fig. 25.53]. Looking at these two examples, it is hard to make it out that the boxes are folded from it. They said that they found out through trial and error without using a computer.) An example found by me is shown in Fig. 3.1. This example is one of the easiest to understand (by my subjectivity). Are these polygons “exceptional” ones with only a few? Actually, it is not. In this chapter, we introduce the latest results on nets in which these multiple boxes can be folded. Many interesting nets brought by combinations of mathematics and algorithms appear. If you are good at programming, you may want to make it yourself and make new discoveries.

## References

- [AT08]T. Asano, H. Tanaka, Constant-working space algorithm for connected components labeling, in
*IEICE Technical Report*, vol. COMP2008-1 (2008), pp. 1–8Google Scholar - [BCD+99]T. Biedl, T. Chan, E. Demaine, M. Demaine, A. Lubiw, J.I. Munro, J. Shallit, Notes from the University of Waterloo algorithmic problem session, 8 September 1999Google Scholar
- [DHK+19]M.L. Demaine, R. Hearn, J.S. Ku, R. Uehara, Rectangular unfoldings of polycubes, in
*Canadian Conference on Computational Geometry*(2019), pp. 164–168Google Scholar - [DO07]E.D. Demaine, J. O’Rourke,
*Geometric Folding Algorithms: Linkages, Origami, Polyhedra*(Cambridge University Press, Cambridge, 2007)Google Scholar - [Gar08]M. Gardner,
*Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner’s First Book of Mathematical Puzzles and Games*(Cambridge University Press, Cambridge, 2008)Google Scholar - [Gol94]S.W. Golomb,
*Polyominoes*(Princeton University Press, Princeton, 1994)Google Scholar - [KRU07]M. Kano, M.-J.P. Ruiz, J. Urrutia, Jin Akiyama: a friend and his mathematics. Graphs Comb.
**23**(Suppl), 1–39 (2007)Google Scholar - [MK06]G. Miller (D.E. Knuth), Cubigami,
*Cubism for Fun*, vol. 70 (2006), pp. 24–27, http://www.puzzlepalace.com/ - [MHU19]K. Mizunashi, T. Horiyama, R. Uehara, Efficient algorithm for box folding, in
*International Conference and Workshop on Algorithms and Computation (WALCOM 2019)*. Lecture Notes in Computer Science, vol. 11355 (2019), pp. 277–288Google Scholar - [Xu17]D. Xu, Research on developments of polycubes, Ph.D. thesis, Japan Advanced Institute of Science and Technology (2017)Google Scholar
- [XHS+17]D. Xu, T. Horiyama, T. Shirakawa, R. Uehara, Common developments of three incongruent boxes of area 30. Comput. Geom.: Theory Appl.
**64**, 1–17 (2017). https://doi.org/10.1016/j.comgeo.2017.03.001